Bicomplex Algebraic Numbers

TL;DR

This paper extends classical algebraic number theory concepts to bicomplex numbers, introducing new types of extensions, decomposition properties, and examples, thus broadening the understanding of algebraic structures in bicomplex settings.

Contribution

It generalizes the primitive element theorem to bicomplex extensions and establishes fundamental decomposition properties for these extensions.

Findings

Primitive element theorem generalization for bicomplex extensions

Decomposition property for bicomplex extensions

Examples of finite bicomplex extensions

Abstract

We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct classes of extensions depending on the nature of the generating element. We further establish a key decomposition property for bicomplex extensions, which serves as a foundation for studying their rings of integers. We also observe that prime elements in the ring of integers of a number field may become semiprime in the rings of integers of suitable bicomplex extensions. Finally, we present two explicit examples of finite bicomplex extensions.